Harold Williams interview

Harold Williams (1912-1981) taught in the social sciences department at Central Washington College of Education (predecessor to Central Washington University) from 1948 to the 1970s.https://digitalcommons.cwu.edu/cwura_interviews/1093/thumbnail.jp

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.Comment: 47 page

Toda Systems, Cluster Characters, and Spectral Networks

We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster coordinates defined via spectral networks, we identify the phase space of this system with the wild character variety related to the periodic nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We show that this identification takes the relativistic Toda Hamiltonians to traces of holonomies around a simple closed curve. In particular, this provides nontrivial examples of cluster coordinates on $SL_n$-character varieties for $n > 2$ where canonical functions associated to simple closed curves can be computed in terms of quivers with potential, extending known results in the $SL_2$ case.Comment: 37 pages; Minor updates from previous versio